Projectile motion: determine launch angle

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  • #1

Homework Statement



A cannon is supported one meter above the top of a 20 degree declining slope of length 200m. The cannon has a launch velocity of 55m/s. There is a ball halfway down this slope moving at a constant velocity of 20m/s.

-Determine the angle [itex]\Theta[/itex] required for the projectile to hit the bottom of the hill within one degree.
-How long does the projectile take to hit the bottom of the hill?
-There is a target moving at a constant velocity of 20m/s down the hill. How long should you delay firing to hit the target as it reaches the bottom of the hill?
-Find the maximum distance this projectile can travel as measured from the bottom of the hill.


Homework Equations



x=x[itex]_{0}[/itex]+v[itex]_{0x}[/itex]t
v[itex]_{y}[/itex]=v[itex]_{0y}[/itex]t-gt[itex]^{2}[/itex]
y=y[itex]_{0}[/itex]+v[itex]_{0y}[/itex]t-0.5gt[itex]^{2}[/itex]
v[itex]^{2}_{y}[/itex]=v[itex]^{2}_{0y}[/itex]-2g(y-y[itex]_{0}[/itex])


The Attempt at a Solution



I have tried using all of the two-dimensional kinematics equations with some success but the problem is that [itex]\Theta[/itex] is unknown so I can't find t or v[itex]_{fy}[/itex].

My best attempts are as so:
v[itex]_{yf}[/itex]=55sin[itex]\Theta[/itex]-9.81t
188=55cos([itex]\Theta[/itex])t
188=0+55cos([itex]\Theta[/itex])t-4.9t[itex]^{2}[/itex]

The diagram for the problem is attached.
Either it is algebra more complicated than I am used to or I am missing something very simple. I am confident that if I could solve for [itex]\Theta[/itex] OR t I could finish the problem easily. I can't seem to get a solveable equation. Thank you for your help and I apologize if there are syntax errors as I am a first time user.
 

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Answers and Replies

  • #2
35
2
There is inconsistency , the questions on the attachment and the problem statement you provided . I would like to help you but there are some ambiguity .
 
  • #3
I apologize for that. I just updated the post to match the wording of the diagram exactly. I have continued to work on this problem and used a few pages of paper, but still have not been able to find a way to solve for theta.
 
  • #4
35
2
You have to find the range in terms of the initial angle then try to optimize it . That is how I solved it . You have to think why one needs an optimization approach to solve this problem.
 

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